See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/248532186 The fractal nature of geochemical landscapes ARTICLE in JOURNAL OF GEOCHEMICAL EXPLORATION · APRIL 1992 Impact Factor: 2.75 · DOI: 10.1016/0375-6742(92)90001-O CITATIONS 86 READS 20 4 AUTHORS, INCLUDING: J. Feder University of Oslo 165 PUBLICATIONS 5,855 CITATIONS SEE PROFILE Torstein Jøssang University of Oslo 115 PUBLICATIONS 3,433 CITATIONS SEE PROFILE Available from: J. Feder Retrieved on: 04 February 2016
B. B/51viken a, P.R. S t o k k e b, j . Feder c and T. J6ssang c a Geological Survey of Norway, P.O. Box 3006 Lade, N-7002 Trondheim, Norway
b Statoil, P.O. Box 300 Forus, N-4001 Stavanger, Norway c Department of Physics, University of Oslo, P.O. Box 1048, 0316 Oslo 3, Norway
(Received 21 November 1989; accepted after revision 31 October 1991 )
ABSTRACT
B61viken, B., Stokke, P.R., Feder, J. and J6ssang, T., 1992. The fractal nature of geochemical land- scapes. J. Geochem. Explor., 43: 91-109.
Fractals are shapes that look basically the same on all scales of magnification - - they are self-like. Numerous natural phenomena have this property, and fractal geometry has contributed significantly to their analysis. Geochemical maps and other geochemical data from the literature indicate that geochemical dispersion patterns (geochemical landscapes ) may have fractal dimensions because they appear similar at all scales of magnitude from microscopic to continental, in agreement with diverse geological processes of varying rapidity and spatial extent ranging from chemical reactions to conti- nental movements.
Analysis of variogrammes and other tests carried out on geochemical dispersion patterns (contents of 21 acid soluble elements in 6000 samples of stream sediment) within a 250,000 km 2 survey area in northern Fennoscandia indicates a fractal dimension of between 2.1 and 2.9 for 10 elements (AI, Ba, Ca, Fe, Li, Mg, Sc, Sr, V and Zn), while the remaining 11 (Ag, Ce, Co, Cr, Cu, La, Mn, Mo, Ni, P and Zr) give inconclusive results presumably due, mainly, to inadequate precision of the chemical analyses. These fractal dimensions were found to exist between distances of 5 and 150 kin, which are the linear limits set by the sample spacing and the size of the survey area respectively.
It is proposed that various sets of geochemical, geophysical and other types of data from around the world be analyzed for their fractal properties, collecting evidence as to whether fractal dimensions are a general quality of geochemical dispersion patterns. If it can be shown that geochemical landscapes are usually true fractals analogoues to topographic landscapes, the impact on applied geochemistry may be profound. One practical consequence in mineral exploration could be the possible existence of numerous economically interesting regional to continental geochemical provinces on earth. Such provinces could be detected at relatively low cost through analysis of wide-spread samples. Consecu- tive denser sampling within the disclosed provinces would reveal subprovinces, which again could be investigated further by successively more intensive sampling. This type of systematic survey employ- ing the principles of fractal geometry for stepwise selection and progressively more thorough exami- nation of subareas of decreasing size, would apply to any survey area of interest and could improve cost-efficiency in mineral exploration.
I N T R O D U C T I O N
The "fractal geometry of nature" (Mandelbrot, 1983) is a new develop- ment with numerous applications in science. The concept is in process of being
*Correspondence to: B. B61viken, Geological Survey of Norway, P.O. Box 3006 Lade, N-7002 Trondheim, Norway.
Examples of processes causing geochemical dispersion patterns at various scales. Most of the pro- cesses take place under addition of energy (solar or terrestrial ) and promote heterogeneity or homo- geneity depending on the circumstances. Several of the processes span over much greater intervals than those indicated
Linear scale Examples of processes (m)
<10 ~
10 6-10 2
10-2-102
102-106
>106
Chemical reactions, melting, solidification, diffusion, dissolution
Ground water movement, soil formation, plant growth, down slope movements, igneous activity, erosion, sedimentation, meteoritic impacts
Volcanic eruptions, mountain making, transport in surface water. Glacial displacement, seafloor spreading, animal life processes
Continental movements, river water transport, oceanic currents, atmospheric transport
TABLE 2
Examples of types of geochemical landscapes (geochemical dispersion patterns) at various scales
Linear scale Examples of distribution patterns (m)
<10 6
10-6_10 2
10-2_102
102-106
>106
Distribution of trace elements in minerals as imaged by electron microprobe back-scatter X-rays (Fig. 1 )
Distribution of minerals in thin sections
Distribution of the contents of major and minor elements in drill cores ( Fig. 2 )
Local to countrywide, distributions of the contents of major and minor elements in geological samples
Metallogenic and geochemical provinces at countrywide to continental scales (Fig. 3)
recognized also in geosciences (e.g. Burrough, 1981; Plotnick, 1986; Tur- cotte, 1986; Chiles, 1988; Robert, 1988; Yfantis et al., 1988; Scholz and Man- delbrot, 1989), and it would appear that the application in geochemical ex- ploration warrants special attention.
Fractals are shapes that look basically the same on various scales of mag- nification. They are neither entirely regular, nor entirely random and it may
THE FRACTAL NATURE OF GEOCHEMICAL LANDSCAPES 93
Fig. 1, Electron microprobe backscattered X-ray image of the contents of barium (white spots) in a sample of iron and manganese nodules taken in a stream at UTM coordinates 526.60X and 6731.00Y, Map sheet 1716 III Vassfaret, Norway. After Kjeldsen (1986).
be said that fractal objects are "self-like" or self-similar. This self-similarity is a type of symmetry that can be used to characterize disordered geometric systems. The methods of fractal geometry can also be used to analyze non- geometrical objects, for example distributions in space or time of various types of observations or measured values. Objects or phenomena that are created by stochastic processes where all length scales have equal opportunity to be realized, become fractal.
Geochemical distribution patterns (geochemical landscapes) may well be- long to this type of scale independent phenomena, since the processes causing such patterns have occurred throughout the geological history of the earth, at any speed and at scales ranging from microscopic to the size of continents (see Table 1 ). Moreover, a vast amount of empirical data shows that signifi-
Higher Grade Minerat~ion (>3 grade units) and Drill tntercep~
Lower Grade MinemJizmion (1-3 grade units) and Drill Intercepts I I I I
Ve~' Lo, Grade (<1 grade unit), o~ No Mineralization of Type Sho~, and Drill Intercepts 0 meters 150
Fig. 2. Lateral distribution of platinum group elements in drill holes from basal and foot wall zones of the Minneapolis Adit, Stillwater, Montana. The grade units given are proportional to the measured grades. After Raedeke and Vian (1986).
cant geochemical distribution patterns exist at a wide range of scales. Classes of such patterns are listed in Table 2 and illustrated by examples in Figs. 1-3.
Whether geochemical distribution patterns of certain elements in various sampling media can be usefully characterized by fractal dimensions must be tested on empirical data sets. We report on an introductory study of this type using stream sediment data obtained during the "Nordkalot t Project" (B61- viken et al., 1986, 1990) within a 250,000 km 2 survey area in northern Fen- noscandia. For readers that are not familiar with fractals, we first present a short general summary of the concept and a brief description of some meth- ods for testing of fractal dimensions. Those who would like a more in-depth treatment of these subjects are referred to the rapidly increasing literature on fractals (see for example Feder, 1988 and Scholz and Mandelbrot, 1989 ). For exploration geochemists, the practical consequences are of particular interest and these will be discussed towards the end of the paper.
F R A C T A L G E O M E T R Y
Comprehension of the concept "fractal" can be attained through a well known example (Mandelbrot , 1983; Feder, 1988). If we study the problem
T H E F R A C T A L N A T U R E O F G E O C H E M I C A L L A N D S C A P E S 95
• , - • .**
Ba ( p p m ) " • , . . ~ . : . : - ~ . .
• • • o , ° *
" : • co ,o%q , "; i i'*~.. 190 . . 6 * o o . o .(!
Fig. 3. C o n t e n t s o f aqua regia soluble Ba in the minus 0.06 m m f rac t ion o f 1057 composite samples of till, Finland. (After Koljonen et al., 1989. ) The east-south crossing zone of high Ba concentrations coincides with the Main Sulphide Belt of Finland ( K a h m a , 1978), w h i c h con- t a i n s some of the country's most important economic sulphide deposits.
"How long is the coastline o f Norway," we arrive at the surprising conclusion that the coast o f Norway, as well as other coastlines in the world, have no definite length; the estimated distances depends on the unit used to measure the coastline. Furthermore, the coastline does not have dimension D = 1 as expected from Euclidan geometry, but a "fractal dimension", Dz somewhere between that o f a line ( D = 1 ) and that o f an area ( D = 2 ) . Correspondingly,
96 a. BOLVIKEN ET AL.
30
Y 20 o o o
01
L (a ) = a ~ 1-u
D = 1 . 5
0 20 40 60 80 100
YARDSTICK (km)
Fig. 4. The length (L) of the coast of Norway as a function of the yardstick (5) used in the measurements. The equation L ( 5 ) = a5 ~- o, can be fitted to the line for a value of Dt = 1.5. where a is a proportionality factor dependent on the units used.
for fractal areas DA is between 2 and 3 and for fractal volumes D~. is above 3. These are typical properties of fractal objects - - they have a fractal dimension.
We can estimate the length of the coast o f southern Norway by "walking" the yardstick, which has a length of let us say 50 = 100 kin, along the mainland coast. The number of steps needed to go from one end to the other, always stepping on the coastline, is N(5o). The length estimate L0 = N(5o)50 clearly underestimates the length of the coastline since even rather large t]ords and bays do not contribute to this coarse measurement. Repeating the procedure using a smaller yardstick, 5|, yields a larger estimate L~ =N(S i )5! for the length of the coastline. At increasingly higher resolutions more and more details are captured, and the measured length increases with decreasing yardstick as shown by the curve in Fig. 4. The equation
L (d ) = a S ' - o , ( 1 )
can be fitted to this curve. Here L ( 5 ) is the length of the coastline measured with a certain yardstick, & and a is a factor of proportionali ty that depends on the units used. DI can be easily calculated by computing the logarithms of eq. 1, obtaining a linear equation. In this case Dt is 1.5, which means that the coast of Norway is fractal with the effective fractal dimension Dz- 1.5.
The simple power-law function of eq. 1 is indispensable for curves that "look the same" at any magnification - - it implies self-similarity or scaling, i.e., invariance under changing length scales. For an ordinary Euclidean curve one would expect L (5) to be independent o f & This can be obtained only if Dt = 1, indicating the dimension 1 of Euclidean curves.
The coastline of Norway is an example of a natural fractal. A well-known example of a mathematical fractal is the Koch curve, which has the dimen-
THE FRACTAL NATURE OF GEOCHEMICAL LANDSCAPES 97
Fig. 5. Construction of the Triadic Koch curve. After a sufficient number of iterations the curve looks the same at all scales.
sion Dz=ln4/ ln3 = 1.2618 (Fig. 5 ) and is shown here for the purpose of ex- plaining the principles.
DESCRIPTION OF THE GEOCHEMICAL DATA-SET
The geochemistry sub-project of the Nordkalot t Project has been described by B/Slviken et al. (1986, 1990). Within each 30 km 2 of the 250,000 km 2 sur- vey region one drainage area (approximately 10 km 2 in size) was selected for sampling. A sampling station was located at the downstream end of each drainage area. Stream sediment was collected at those 5773 sites where sam- pling was possible. The samples were taken at 5-10 subsites within a 50 m section of the stream and wet-sieved to grain-size fraction < 0.18 mm. This fine fraction was digested 3 h with 5 ml hot ( 110 °C) 7 N HNO3. After dilu- tion to 20 ml and centrifugating, 21 elements (Ag, A1, Ba, Ca, Ce, Co, Cr, Cu, Fe, La, Li, Mg, Mn, Mo, Ni, P, Sc, Sr, V, Zn, Zr ) were determined by induc-
98 I~. BOLVIKEN ET AL.
Fig. 6. Contents of nitric acid soluble AI in the minus 0.18 mm fraction of stream sediment from northern Fennoscandia (see also Fig. 7).
tively coupled, argon-plasma spectrometry (Odegard, 1981). The obtained concentrat ions were plot ted on maps using dots of varying size to indicate the analytical result (Bj/Srklund and Gustavsson, 1987 ). An example of the maps obtained (A1) taken from B/51viken et al. (1986) is given in Fig. 6 and the corresponding frequency distr ibut ion in Fig. 7.
METHODS FOR TESTING OF FRACTAL DIMENSIONS
We have applied four of the available methods for testing fractal dimen- sions of geochemical dispersion pat terns (geochemical landscapes ): ( 1 ) analysis of variograms (2) length of contour versus measur ing yardstick
THE FRACTAL NATURE OF GEOCHEMICAL LANDSCAPES 99
=~ 5 - t ~ • * * t + t t
0.3 0.5 0.7 1 2 3 4 AI % Fig. 7. Cumulative frequency distribution diagram for the contents of acid soluble AI in the minus 0.18 mm fraction of stream sediments from northern Fennoscandia, see Fig. 6. Note the logarithmic scale along the abscissa. -y
C
O ~ - h 0 Fig. 8. The "ideal" shape semi-variogram. (h) Distance between sample pairs; (y) variance of increments:
2 y ( h ) = ( g ( x i ) - g ( x i + h ) ) 2 r l i = l
(g) concentration of a given chemical element; (x) sample locality; (c) sill reached at distance h=a.
( 3 ) the area-per imeter relation (4) the number -a rea relation O f these, the analysis of variograms is most straightforward, because raw data can be used in testing. The other three methods are less direct because they require interpolation of geochemical concentrations between sample points to obtain isoconcentration contours. A widely accepted method (Kriging) is available for such interpolation (see for example, Clark, 1979). From our data-sets of irregularly spaced geochemical samples, we have est imated values in a square grid by Kriging. Linear dimensions of unit cells were chosen to be 1 km × 1 km; i.e., 18% of the original average sampling interval. For each point on the grid we solved the Kriging equations, including all measured concentrations within a circle of 50 km. This typically gave 65 independent linear equations for each est imated concentration. The Kriging process re-
1 O0 B. B(3LVIKEN ET AL.
suits in smoothing of the iso-concentration contours; and the fractal dimen- sions, D, obtained from Kriged data, therefore, underestimate the real values.
Analysis of variograms
A semi-variogram (Fig. 8 ) is a graphical representation of how the average variance of pairs of geochemical concentrations (variance of increments, 7(h) ) varies with distance, h, between the samples in the pair. This variance is expressed mathematically by
1 n 27(h) = n ~ (g(x,) -g(x, + h) )2 (2)
where n is the number of all possible sample pairs at a given distance h be- tween samples, g(x, ) is the element concentration at point x,, and g(x,+ h) is the element concentration at distance h from point xi. The factor 2 is intro- duced following a common convention.
A fractal distribution of geochemical concentrations implies that the vari- ance of increments would increase perpetually with increasing distance, h, between sample pairs (within the theoretical limit of component concentra- tions totalling no more than 100%). This steady increase for all values of h is a consequence of the self-similarity of fractal patterns at all scales.
For a fractal surface, the semi-variogram will follow the equation (Man- delbrot, 1983, p. 353; Feder, 1988, p. 204):
y ( h ) = ( h / 2 ) 2H (3)
where h is the distance between samples in a sample pair, 2 is a constant, and H is the Hurst exponent, which is related to the fractal dimension DA by the equation
H= 3-D.,
The fractal dimension (DA) can, therefore, be estimated from a log-log plot ofeq. 3.
However, the "ideal" semivariogram (Fig. 8, Clark, 1979 ) has a horizontal sill, appearing to conflict with the idea that variance increases with the dis- tance, h, between sample points for all values of h.
This discrepancy is, however, not real for two reasons: ( 1 ) For any natural fractal phenomenon there is a lower and an upper practical length or scale for self-similarity. In the case of geochemical dispersion patterns obtained within a certain area, the lower length scale is the sampling interval and the upper length scale is the linear dimension of the survey area. Outside these limits the empirical geochemical landscape is mathematically nonexistent; the var- iance, consequently approaches a constant value as h approaches the linear dimension of the survey area. (2) The "ideal" variogram presupposes normal
THE FRACTAL NATURE OF GEOCHEMICAL LANDSCAPES 101
(or near normal) distribution of the data. For log-normal type distributions (which is common for trace elements), the variogram may have the "ideal" shape even though the data are fractal and well within the lower and upper practical limits of sampling. Log-normal distributions have a great range of concentrations and only variations in high concentrations will contribute sig- nificantly to the sum of the variance of increments; although the relative vari- ation may be great at low concentrations. In such cases, log transformation of concentrations before calculating variances would produce a variogram with- out a sill, even for log-normal distributions. However, log transformation is rarely acceptable in this connection because the variance become enlarged by possible poor precision of the chemical analyses at low concentrations, which may result in too much noise for detecting real variations at higher concentrations.
These considerations lead to the conclusion that untransformed data should in most cases be used for computing variograms when testing fractal dimen- sions of geochemical landscapes. However, the fractal dimension, D, tends to be overestimated when employing untransformed log-normal data, and the results should be treated with care.
Length of contour versus measuring yardstick
This method is described in the Introduction, see eq. 1 and p. 96.
The area-perimeter relation
A fractal dimension of a landscape requires that the relation between the perimeter of an area (here, length of geochemical iso-concentration contour) and the area within the perimeter (here, area inside a geochemical contour) follows the equation (Feder, 1988, pp. 200-201 ):
P= CA 0,/2 (4)
where A is the area of the island (closed contour), P is the perimeter of A, C is a constant and Dt is the fractal dimension of the coastline.
This method of testing for fractal dimensions is advantageous because se- lection of coastlines (contours) can be adjusted according to the sensitivity of the analytical method, thus avoiding substantial errors due to poor analyt- ical precision at low concentrations. However, the data-set should be square gridded for computer treatment, and the required interpolation between ir- regularily spaced sample points would be a drawback as mentioned above.
102 B. BOLVIKEN ET AL.
The number-area relation
If all islands of a region are listed by decreasing size, and a is a possible value for the area of an island, the following number area relation for fractal landscapes is found (Mandelbrot, 1983, p. 118).
Nr(A>a)=F'a -8 (5)
where Nr (A > a ) is the total number of islands with size A above a, and F and B are positive constants. Mandelbrot (1983) argues that for mutually similar islands, B should equal Dr~2, where D~ is the fractal dimension of the coast- line. The previous remarks regarding adjustment to the sensitivity of the an- alytical method, as well as the requirement for interpolation between data points for the area-perimeter relation, are also valid in this case.
R E S U L T S
Of the 21 elements tested, 11 (Ag, Ce, Co, Cr, Cu, La, Mn, Mo, Ni, P, and Zr) produced variograms that did not fit with eq. 3, and variograms are pre- sented for the remaining 10 (A1, Ba, Ca, Fe, Li, Mg, Sc, Sr, V and Zn, see Fig. 9 ). For the other methods, results for aluminium are given as examples ( Figs. 10-12).
The reason for misfit of some of the variograms with eq. 3 is not necessarily that the true dispersion patterns of these elements are not fractal. For some elements the solubility in HNO3 is low and of doubtful significance. The an- alytical precision is often poor at low concentrations (B/51viken et al., 1986, p. 15 ). Although reproducibility is satisfactory for practical mapping purposes, the utilization of the data in variograms covering the entire concentration range could be inappropriate for some elements.
A1 shows the most unambiguous variogram (Figs. 9, Table 3 ). For AI the curve 7(h)= ( h / 2 ) 2 H fits the data well up to a sill at 150 km, which corre- sponds to the maximum length of the survey area. The Hurst exponent H for A1 is calculated to 0.109, corresponding to a fractal dimension DA -~ 2.9, which is rather high and indicative of a rugged surface. Similar, values for H were obtained for the other elements with acceptable fitting (Ba, Ca, Fe, Li, Mg, Sc, Sr, V and Zn) (Table 4). We conclude that the variogram shapes for these elements are in agreement with fractal scaling of the data.
An example of the total length, L, of the coastlines of all islands of a given A1 concentration, as function of the yardstick or resolution length ~ is given in Fig. 10. For 0.785% A1 and yardstick ~ up to about 10 km, there is a very good fit of the scaling equation 1, with fractal dimension for the coastline given by Dr= 1.21, corresponding to DA = 2.21 for the landscape.
The area-perimeter relation for AI (Fig. 11 and Table 4) fits well with the theoretical curve (eq. 4), and the fractal dimension for the contours of alu-
THE FRACTAL NATURE OF GEOCHEMICAL LANDSCAPES 103
2.00 1.59 AJ
1 . 2 6
1.00 ~ .79
.63
"~x 2.51 t 1.59 c~
~_. 1.00 /
.63
II .40 ¢~ 2,00 it) Li t"
E loo
0
'*~ .50
159 [ 2.00 /
.~ sc
~ 1.28j
~1°° I ~ .79 " E 2.00 r ~_ 1.59 ; v
I 1.26
1 . 0 0
7 9 L 1
f f z ~
i
Ba
' ~ / ~ ' ~ ' ~ - Sr
• 12" 1.59
j l.26 I • 1.00
J j / .:~<....-~:""
2.00
1.59
1.00 2.00
1 . 5 9
1.26
1.00
.79 2.53
1 . 5 9
126
.63
~..~.~-'"~ ...... .
10 30 100 300 1 3 10 30 100
Distance Between Samples (h) km
i 2.00 1.59
1.26
i 1,00 J .79
300
Fig. 9. Normalized semi-variograms (dots) for 10 acid soluble elements in the minus 0.18 mm fraction of stream sediment from northern Fennoscandia. The function 7(h) = (h/2) 2H is fit- ted to the variograms. Note the logarithmic scale of the axes.
minum can be estimated to D~= 1.2 (corresponding to DA=2.2) , indepen- dent of the concentration level chosen for the definition of the islands. This fractal dimension must be a lower limit because of the Kriging which, in ad- dition to smooth contours, produces rather straight, artificial coastlines along the border of the survey area.
The histograms of the number of islands with an area A above a given area a as function of a, and the resulting plot (Fig. 12 ), show that the data can be fitted reasonably well with the number area relation (eq. 5 ). The fractal di- mension for the total coastline (contour) is given by DI--- 1.11 corresponding to DA - 2.1 for the area within the contour (Table 4 ).
Fig. 10. The measured length, L(8) , of all the 0.785% AI contours (arithmetic mean of all Al concentrations in Fig. 6) (Kriged values) as a function of the length 8 of the measuring yards- tick. The straight line fit to the observations has the form logL(8)= loga+ (l-D~)logS. Data from distribution patterns of acid soluble A1 in stream sediment from northern Fennoscandia.
~ 1O4
lo3
(~ 102
~/3 101
~oc i i
loo 102 104 ~ o o
Area Above Contour, 0.785% A1 (kin2)
Fig. 1 1. The length of the contour as a function of the area above the contour for the average contents of acid soluble AI (0.785%) in stream sediment from northern Fennoscandia. The contours are drawn between Kriged values in a square grid. The line represents a fit of the area- perimeter equation (4) , P = CA Da2 with/)1= 1.1 1 + 0.01 and C= 4.
A I = 0.785%
<
300 i
100 0 ~ 30
10 A I =
3
] i i i i ,
100 101 102 103 104 105
Area Above Contour, 0.785% A1 (kin)
Fig. 12. The number of areas above the contour 0.7 8 5% AI (Kriged values) as a function of the area. The dots represent the observations and the line represents the fit of eq. 5 to the observa- tions. Data from acid soluble Al in stream sediment from northern Fennoscandia.
106
SOME IMPLICATIONS OF FRACTAL GEOCHEMISTRY
B. BOLVIKEN ET AL.
It is proposed that various types of geochemical and geophysical data sets from around the world be analyzed in order to determine if fractal scaling is general for geochemical dispersion patterns. Regional geophysical data re- flecting the geochemistry, such as results of airborne radiometric measure- ments, are of particular interest since such datasets normally are obtained in a regular grid and can be so large that interpolation between observation points is not a necessary requirement for the tests.
If it is a general rule that geochemical landscapes are fractal - - and, conse- quently, analogous to topographic landscapes - - new opportunities seem to appear for applied geochemistry. Let us speculate about some of the possible consequences in mineral exploration.
There may exist unknown hierarchies of geochemical background/prov- inces/anomalies on earth (similar to topographic areas of low, elevated and high terrain) at any scale from local to continental. Large provinces of ele- vated trace element contents may have a higher frequency of significant min- eral deposits than geochemical background regions. Conversely, ore deposits generally seem to lie within geochemical provinces (B~ilviken et al., 1990).
Data provided or summarized by e.g. Plant et al. (1988), Koljonen et al.
Fig. 13. Principle sketch showing stepwise geochemical mapping starting with low sampling density in an initial large survey area (A0), and following up of the provinces/anomalies (Ab A2, A3, etc. ) successively by increasingly dense sampling for each step. The corresponding fre- quency distribution curves are shown in Fig. 14.
THE FRACTAL NATURE OF GEOCHEMICAL LANDSCAPES 107
30
10
1.0
0.1
CONCENTRATION LOGARTHMIC SCALE
Fig. 14. Frequency distribution curves of element concentration in an initial survey area Ao and anomalous follow-up areas A ~, A2, A3, etc. found by stepwise geochemical mapping with increas- ing sampling density for each step, see Fig. 13. The abscissa has a logarithmic scale, but is with- out a defined length unit.
(1989), Manheim et al. (1989), Ottesen et al. (1989), B/~lviken et al. (1990), and Duval (1990) indicate that geochemical provinces could be so extensive, yet well defined, that a global survey using very wide spread samples would suffice to reveal them. After such continental provinces have been indicated, more dense sampling within them would be appropriate in order to detect subprovinces. More thorough examination of subprovinces could then be done with even closer sample spacing, and so on. This strategy is generalized in the sketches in Figs. 13 and 14. The initial wide range survey area in Fig. 13 could, depending on the circumstances, be of any size: local, regional, countrywide or global.
In the past the application of geochemical surveys in exploration have gen- erally been based on geological modelling. In such surveys tens of thousands of samples have often been collected initially in order to obtain comprehen- sive coverages of large survey areas in a single step. Principles of fractal ge- ometry indicate that it may be profitable to work more empirically (as if the task were to find the topographic highs within an unknown area). A system- atic step by step procedure with progressively increasing intensity of measure- ments in target areas of successively decreasing size, would be more cost effi- cient than complete surveys of whole areas in one step.
Future possibilities include the utilization of principles of fractal geometry in order to find criteria for a selection of the most favourable provinces/ anomalies detected in low density surveys as well as to estimate the most cost- efficient increase in sampling density per step of follow up investigations.
108 B. BOLVIKEN ET AU
C O N C L U S I O N S
( 1 ) Geochemical maps and other empirical data from the literature show that significant geochemical dispersion patterns exist for a number of ele- ments at scales from microscopic to continental, indicating that "landscapes" formed by natural geochemical concentrations may have fractal scalings.
(2) Variograms of geochemical dispersion patterns for the contents of acid- soluble parts of 21 elements in 6000 samples of stream sediment from a 250,000 km 2 survey area in northern Fennoscandia, were used to estimate if the data are fractal. Areal fractal dimensions, Dn, of 2.9 were found for 10 elements (A1, Ba, Ca, Fe, Li, Mg, Sc, Sr, V, and Zn), while the remaining 11 gave inconclusive results, presumably due to inadequate precision of the chemical analyses.
This value for D is an upper limit since estimation from variograms, leads to positively biased results if the data are log-normal and of a limited preci- sion at low concenctrations. The area-perimeter relation and the number- area relation, which underestimates D, gave estimates of DA-~ 2.1. We con- clude that the data are fractal characterized by dimensions DA in the range 2.1 < DA < 2.9. This dimension was found to exist between 5 and 150 km, which are the limits set by the sampling distance and the size of the survey area.
( 3 ) The fractal properties of geochemical and geophysical data sets (as well as other data describing natural resources such as minerals, ores, hydrocar- bons, etc.) should be investigated to determine if fractal dimensions are a general feature of certain types of geographically distributed data. If geo- chemical landscapes are true fractals similar to topographic landscapes, the impact on the art of applied geochemistry could be profound. Some of the consequences in regard to mineral exploration could be: ( 1 ) the existence of numerous unknown regional to continental geochemical provinces detectable by chemical analysis of a moderate number of samples. Such provinces may host economically profitable deposits more frequently than other areas, (2) improvement of cost efficiency in geochemical prospecting by systematic fol- low-up investigations, working step by step increasing sample density within decreasing target areas selected on the basis of the principles of fractal geometry.
A C K N O W L E D G E M E N T S
The senior author is grateful to Purdue University for the opportunity to stay for a six month period in 1989, during which earlier reports on this in- vestigation were compiled into the present paper. We thank Virginia Ewing and/kse Minde for typing the manuscript, Giovanni Soto for arrangement of figures and James Gardner, Carl Olaf Mathiesen and Sally Skow for correct- ing the English.
THE FRACTAL NATURE OF GEOCHEMICAL LANDSCAPES 109
REFERENCES
B~lviken, B., Bergstr6m, J., Bj6rklund, A., Kontio, M., Lehmuspelto, P., Lindholm, T., Mag- nusson, J., Ottesen, R.T., Steenfelt, A. and Volden, T., 1986. Geochemical Atlas of Northern Fennoscandia, scale 1:4 million. Geological Surveys of Finland, Norway and Sweden, 20 pp., 155 maps.
Bt~lviken, B., Kullerud, G. and Loucks, R.R., 1990. Geochemical and metallogenic provinces: a discussion initiated by results from geochemical mapping across northern Fennoscandia. J. Geochem. Explor., 39: 49-90.
Bj/Srklund, A. and Gustavsson, N., 1987. Visualization of geochemical data on map: New op- tions. J. Geochem. Explor., 29: 89-113.
Burrough, P.A., 1981. Fractal dimensions of landscapes and other environmental data. Nature, 294: 240-242.
Chiles, J.P., 1988. Fractal and geostatistical methods for modelling of a fracture network. Math. Geol., 20: 631-654.
Clark, I., 1979. Practical Geostatistics. Applied Science Publishers Ltd., London, 120 pp. Duval, J.S., 1990. Modern aerial gamma-ray spectrometry and regional potassium map of the
conterminous United States. J. Geochem. Explor., 39: 249-253. Feder, J., 1988. Fractals. Plenum Press, New York, 283 pp. Kahma, A., 1978. The main sulphide belt of Finland between Lake Ladoga and the Bothnian
Bay. Geol. Soc. Finland Bull., 50: 39-43. Kjeldsen, S., 1986. Oxidates as a geohcemical sampling medium in granitic terrain. Geological
Survey of Norway, Open File Report 86.169. Koljonen, T., Gustavsson, N., Noras, P. and Tanskanen, H., 1989. Geochemical Atlas of Fin-
land: preliminary aspects. J. Geochem. Explor., 32:231-242. Mandelbrot, B.B., 1983. The Fractal Geometry of Nature. W.H. Freeman, New York. Manheim, F.T., Lane-Bostwick, L.M., Kirschenbaum, H. and Aruscavage, P.J., 1989. Chemical
maps of ferromanganese encrustations in Pacific Ocean. 28th International Geological Con- gress, Abstracts Vol. 2. U.S. Geological Survey, Washington, DC, p. 359.
Odegard, M., 1981. The use of inductively coupled argon plasma (ICAP) atomic emission spec- troscopy in the analysis of stream sediments. J. Geochem. Explor., 14:119-130.
Ottesen, R.T., Bogen, J., B61viken, B. and Volden, T., 1989. Overbank sediment: a representa- tive sample medium for regional geochemical mapping. In: S.E. Jenness et al. (Editors), Geochemical Exploration 1987. J. Geochem. Explor., 32: 257-277.
Plant, J.A., Hale, M. and Ridgway, J., 1988. Developments in regional geochemistry for mineral exploration. Trans. Inst. Min. Metall. (Sect. B. Appl. Earth Sci. ), 97: B 116-B 140.
Plotnick, R.E., 1986. A fractal model for the distribution of stratigraphic hiatuses. J. Geol., 94: 885-890.
Raedeke, L.D. and Vian, R.W., 1986. A three-dimensional view of mineralization in the Still- water J-M Reef. Econ. Geol., 81:1187-1195.
Robert, A., 1988. Statistical properties of sediment bed profiles in alluvial channels. Math. Geol., 20: 205-225.
Scholz, C.H. and Mandelbrot, B.B. (Editors), 1989. Fractals in Geophysics. Special Issue, Pure Appl. Geophys., 131 (1/2), 313 pp.
Turcotte, D.L., 1986. A fractal approach to the relationship between ore grade and tonnage. Econ. Geol., 81: 1528-1532.
Yfantis, E.A., Flatman, G.T. and Englund, E.J., 1988. Simulation of geological surfaces using fractals. Math. Geol., 20: 667-672.
